On the non-integrability of a fifth order equation with integrable two-body dynamics

TL;DR

This paper investigates a fifth order PDE extending the Camassa-Holm equation, showing it exhibits soliton-like solutions but is ultimately non-integrable due to lack of Hamiltonian structure and integrability properties.

Contribution

The paper demonstrates the non-integrability of a complex fifth order PDE that generalizes an integrable equation, despite its soliton-like solutions and stable pulson dynamics.

Findings

Pulsons are stable and exhibit elastic scattering.

The PDE lacks a Lagrangian or bi-Hamiltonian structure.

Painlevé and Wahlquist-Estabrook analyses indicate non-integrability.

Abstract

We consider the fifth order partial differential equation (PDE) $u_{4x,t}-5u_{xxt}+4u_t+uu_{5x}+2u_xu_{4x}-5uu_{3x}-10u_xu_{xx}+12uu_x=0$, which is a generalization of the integrable Camassa-Holm equation. The fifth order PDE has exact solutions in terms of an arbitrary number of superposed pulsons, with geodesic Hamiltonian dynamics that is known to be integrable in the two-body case N=2. Numerical simulations show that the pulsons are stable, dominate the initial value problem and scatter elastically. These characteristics are reminiscent of solitons in integrable systems. However, after demonstrating the non-existence of a suitable Lagrangian or bi-Hamiltonian structure, and obtaining negative results from Painlev\'{e} analysis and the Wahlquist-Estabrook method, we assert that the fifth order PDE is not integrable.